The Calculus Gallery: Masterpieces from Newton to Lebesgue 6.1.2008 Edition

by William Dunham (Author)

44 ratings

ISBN-13: 978-0691136264

ISBN-10: 0691136262

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More than three centuries after its creation, calculus remains a dazzling intellectual achievement and the gateway into higher mathematics. This book charts its growth and development by sampling from the work of some of its foremost practitioners, beginning with Isaac Newton and Gottfried Wilhelm Leibniz in the late seventeenth century and continuing to Henri Lebesgue at the dawn of the twentieth--mathematicians whose achievements are comparable to those of Bach in music or Shakespeare in literature. William Dunham lucidly presents the definitions, theorems, and proofs. "Students of literature read Shakespeare; students of music listen to Bach," he writes. But this tradition of studying the major works of the "masters" is, if not wholly absent, certainly uncommon in mathematics. This book seeks to redress that situation.

Like a great museum, The Calculus Gallery is filled with masterpieces, among which are Bernoulli's early attack upon the harmonic series (1689), Euler's brilliant approximation of pi (1779), Cauchy's classic proof of the fundamental theorem of calculus (1823), Weierstrass's mind-boggling counterexample (1872), and Baire's original "category theorem" (1899). Collectively, these selections document the evolution of calculus from a powerful but logically chaotic subject into one whose foundations are thorough, rigorous, and unflinching--a story of genius triumphing over some of the toughest, most subtle problems imaginable.

Anyone who has studied and enjoyed calculus will discover in these pages the sheer excitement each mathematician must have felt when pushing into the unknown. In touring The Calculus Gallery, we can see how it all came to be.

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Editorial Reviews

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"[A] brilliant book. . . . I predict that Dunham's book will itself come to be considered a masterpiece in its field."---Victor J. Katz, American Scientist

"A joy to read, The Calculus Gallery showcases one of the great intellectual pursuits of all time and, in the words of John von Neumann, 'the first achievement of modern mathematics.' Thirteen scholars, beginning with Newton and Leibniz, who gave birth to calculus in the seventeenth century, are featured in this sequential development of the important ideas that shaped calculus as we know it and gave rise to modern analysis. . . . [I]t is a lovely and engaging gallery of the 'masters' that belongs in the library of everyone who seriously teaches or studies the subject."---Diane M. Spresser, Mathematics Teacher

"A fascinating, competent visit too the calculus gallery."---Eberhard Knobloch, Zentralblatt MATH

"One of Choice's Outstanding Academic Titles for 2005"

"What distinguishes this selection is it truly provides a history of mathematics, not just a history of mathematicians. . . . If a better historical treatment of the development of the calculus is available, this reviewer has yet to see it. . . . Essential.", Choice

"William Dunham, Winner of the 2006 Lester R. Ford Award for his expository article on "Touring the Calculus Gallery" in The American Mathematical Monthly"

"The Calculus Gallery is a wonderful book. The style is inviting; the explanations are clear and accessible. . . . Mathematicians, scientists, and historians alike can learn much that is interesting, much that is mathematically significant, and a good deal that is both."---Judith V. Grabiner, Science

Review

"Pedagogically excellent and extremely well written, The Calculus Gallery bridges the gap between general histories and detailed studies of individual mathematicians. Dunham has described mathematical developments in an engaging style rarely found in literature of this kind."―Annette Imhausen, Trinity Hall, Cambridge

From the Back Cover

"The Calculus Gallery is one of the best efforts at mathematical exposition I have ever read! Dunham presents in detail and in his own words the sequence of ideas of classical giants of mathematics, but each new idea is described in modern terms and notation, so I had absolutely no trouble following along. Furthermore--and this is an astounding achievement--the entire work has a tightly woven development. If it were a detective story I would say it had a plot with no loose ends. An amazing feat. I wish I could plan a single lecture, never mind a course or a book, that well!"--Henry Pollak, Teachers College, Columbia University

"What a fine resource! All of the famous functions that have shaped calculus and analysis parade before the reader in the original words of their creators. Bill Dunham has produced an excellent volume that teachers and students will enjoy and appreciate."--Thomas Banchoff, Brown University

"Bill Dunham has done it again. The Calculus Gallery is a masterly journey through the works of thirteen mathematicians who formulated, formalised, and reformed the calculus into the modern analysis we learn today. Readers of his earlier books have learned to expect a clarity of exposition that few others can attain: they will not be disappointed."--Robin Wilson, author of Four Colors Suffice

"This is an excellent book--an amazing mathematical page-turner. William Dunham has done the seemingly impossible: he has taken some difficult, advanced mathematics and, without sacrificing the technical details, written a lively, readable book about it."--Barry Cipra, author of Misteaks . . . and How to Find Them Before the Teacher Does

"A welcome addition to the literature. The idea of presenting a 'museum of mathematics' is new. It allows the author to present a nonstandard selection of theorems, so that even mathematicians with a strong historical background will learn a few things."--Franz Lemmermeyer, Bilkent University, author of Reciprocity Laws: From Euler to Eisenstein

About the Author

William Dunham, Koehler Professor of Mathematics at Muhlenberg College, is the author of Journey Through Genius: The Great Theorems of Mathematics; The Mathematical Universe; and Euler: The Master of Us All. He has received the Mathematical Association of America's George Pólya, Trevor Evans, and Lester R. Ford awards, as well as its Beckenbach Prize for expository writing.

Publisher : Princeton University Press; 6.1.2008 edition (July 21, 2008)
Language: : English
Paperback : 256 pages
ISBN-10 : 0691136262
ISBN-13 : 978-0691136264
Item Weight : 13 ounces
Dimensions : 5.75 x 0.75 x 9 inches

Best Sellers Rank: #1,039,728 in Books (See Top 100 in Books)
- #752 in Calculus (Books)
- #956 in Mathematics History

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Shum

Great highlighting of key issues for cleaning up the foundations of the calculus

Reviewed in the United States on February 22, 2015

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Shows the power of counterexamples in mathematical development and a great reminder of the key issues calculus learners would have encountered if they were lucky enough to have had good teachers. If you didn't, this book would open a whole new world of mathematical thinking to you. A great book for teachers of calculus at all levels and an exciting story of the calculus for all exposed to the ideas of the calculus now or in their past. For me, as a former researcher of the role of counterexamples in the development of mathematical thinking, this book highlights the role counterexamples played in mathematicians thinking about the key problems and solutions in moving from quick computations to the real subject of real analysis. It is a great story of mystery and resolution, missing in the exposure to the calculus for many learners. If you are not so mathematically inclined, you can gloss over some of the details and see the key ideas and mysteries. If you have a mathematical bent, it is also exciting to see how great mathematicians thought as they grappled with the big ideas of calculus and analysis. It is a very hard book to put down, even to eat or sleep — a great read!

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vijay

Amazing book. I have a degree in Electrical Enginnering ...

Reviewed in the United States on March 31, 2018

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Amazing book.
I have a degree in Electrical Enginnering and have done courses in basic calculus, ordinary and partial differential equations, and several other subjects that depend on calculus. I thought I understood calculus - turns out I was wrong. This book covers the early history of calculus - including the works of Newton, Euler and the Bernoulli brothers. It then moves on to Cauchy, Riemann, Weierstrass (who put the subject on a solid theoretical foundation), and ends with a section on the more "modern" mathematicians - Cantor, Lebesgue and Baire.
A background in mathematics (advanced high school or college) is desirable but not essential. The author keeps the mathematical results and proofs clean and easy to follow. I recommend following the proofs with a pen and paper - it is tough going but has its rewards. The book has changed the way I look at calculus - I am inspired to find and take a course calculus on Youtube again.

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MarcTop Contributor: Swimming

Dunham at his best, must have for the student

Reviewed in the United States on February 3, 2012

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Any Dunham book is worth a read, and I cannot pick a favorite. So I know how he must feel choosing which great theorems to put in a book like this.

This book does require a basic understanding of Calculus, though he simplifies the theorems so that a student will find it useful, but it is not a course text.
In his foreward he mentions this is not a historical book, but in some ways it is. He almost develops the story of Calculus, told through some of the greatest contributors to analysis.

He puts the theorems in a historical context and lets the story unfold. Where this is extremely useful for students is that there is a context beyond a sidebar in a textbook. The reader will find some of the same examples in their textbook, but not presented in the context as Dunham.

There are few authors who do this well, and Dunham is one of the greats.

6 people found this helpful

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Nicholas A. Fortis

Very Good IF you have a solid background

Reviewed in the United States on February 11, 2007

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And by "solid background", I mean a good understanding of honors-level algebra (high school senior variety) AND a reasonably thorough understanding of basic calculus notions such as limits, integration, and differentiation. Even if your background is not quite this strong, I feel you will understand parts of the Newton Leibnitz Euler chapters, but you may start to struggle a touch after that. And, as was pointed out in an earlier review, this book is NOT for the General Reader. Very far from it, in spite of professor Dunham's substantial skill as a science/math writer. (A pleasure to read the gentleman!)

In fine: I studied applied math at UCLA some 55 years ago, and I have not done any serious math since about 1970, so I may be overstating the requirements necessary to enjoy this work, given my own very dim memory of some key elements of mathematics and of the mathematical proof. But for those who feel qualified, this is yet another wonderful piece of work by a truly gifted author.

Naf February 2007; Los Altos CA

32 people found this helpful

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Carles V

A wonderful book

Reviewed in the United States on January 28, 2013

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William Dunham is one of the best "mathematics expository" writers I've ever found.

In this book, Dunham explains the highlights of Theory of Calculus, from the seventeenth century (with Newton and Leibniz) until the dawn of the twentieth (with Cantor and Lebesgue).

Dunham's style is, as usual, brilliant, clear, accesible, exciting, interesting, engaging, motivating, etc.

He talks about some of the main results of the following mathematicians: Newton, Leibniz, The Bernouillis, Euler, Cauchy, Riemann, Liouville, Weierstrass, Cantor, Volterra, Baire and Lebesgue.

From this sequence of masters, the reader can follow the growth and development of calculus and realise how the main concepts of Calculus began in a very intuitive (geometric) way, sometimes even with errors or lacking of rigor and with the work of one generation of mathematicians after the other, they were gradually improved, arriving at the modern ideas we can find in any textbook.

Liouville's section is specially brilliant, with his finding of the world's first transcendental number.

Everyone who seriously teaches or studies calculus must read this book.

4 people found this helpful

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Amazon Customer

Great fun!

Reviewed in the United States on February 25, 2014

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If you are into math (and have even a rudimentary background in calculus) this is an awesome
look at the development of real analysis. Unlike extremely boring puff pieces that glorify this mathematician or that physicist, this
book focuses on concrete ideas and lets them speak for themselves.
Dunham develops the material in a systematic way while still managing to convey how ingenious and cool it all is. I had a wonderful time working through the proofs while reading the text. This is one of the best math books (beginner, advanced, recreational) that I have ever read.

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Never the Twain

A gallery with some masters and a lot of empty spaces

Reviewed in the United Kingdom on August 13, 2010

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If you are reading this review you probably have an interest in Mathematics and Calculus. If you do, then buy this book. It gives an interesting account of the history of Calculus and of the characters that have refined and developed it over the course of centuries. The book glosses over some aspects with the odd note implying that modern mathematicians would be horrified if anyone used such a technique today, but it does not offer any insight into these developments (e.g. the handling of discontinuities) in the modern age.

If however you are a mathematician looking for a good overview of Calculus techniques then this is definitely not it, with the emphasis very much on the personalities and the context of their work.

Antonio

Little detail in some of the proofs

Reviewed in the United Kingdom on April 17, 2014

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By chance, the proofs I was most interested in didn't show up in the book. But it is an excellent book.

A. Lopez

Buen libro

Reviewed in Spain on March 17, 2018

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Buen libro para leer si te gustan las matemáticas y estás interesado en la historia. Para mi gusto, peca algo de excesivamente teórico (aunque para seguramente para un matemática sea una delicia). Muy interesante.

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Cliente Amazon

Una obra de arte

Reviewed in Spain on January 27, 2017

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Pequeñas obras de arte cuya lectura te pone en la piel de algunos de los grandes matemáticos que hicieron la historia del ''nuevo'' Cálculo (Newton, Leibniz,Cauchy, Riemann, Lebesgue.....)
pinceladas geniales para los amantes de la historia de la matemática. Por qué nadie tradujo este libro al español?

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Ronald Mullin

Dunham is a skilled and knowledgable author who chooses interesting ...

Reviewed in Canada on May 20, 2016

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Dunham is a skilled and knowledgable author who chooses interesting parts of the developments of Calculus as it evolves from dubious foundations to the firm foundations used at the present, The reader encounters several surprises.

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The Calculus Gallery: Masterpieces from Newton to Lebesgue 6.1.2008 Edition

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